1. Field of the Invention
The invention relates to the field of electrical power systems, and more particularly to methods for on-line transient stability analysis, on-line dynamic security assessments and energy margin calculations of practical power systems.
2. Description of the Related Art
Power systems are continually experiencing disturbances. These disturbances can be classified as either event disturbances or load disturbances. Power systems are planned and operated to withstand the occurrence of certain disturbances. At present, modern energy management systems (EMS) periodically perform the task of on-line (static) security assessment to ensure the ability of the power system to withstand credible contingencies (disturbances.) The set of credible contingencies is a collection of disturbances that are likely to occur with potentially serious consequences. The assessment involves the selection of a set of credible contingencies followed by the evaluation of the system's ability to withstand their impacts.
The extension of EMS to include on-line dynamic security assessment (DSA) is desirable and is becoming a necessity for modern power systems. This extension, however, is a rather challenging task; despite the consistent pressure for such an extension, partly due to economic incentives and partly due to environmental concerns, performing DSA has long remained an off-line activity. Several significant benefits can be expected from this extension. First, power systems may be operated with operating margins reduced by a factor of 10 or more if on-line, rather than off-line, DSA is performed. A second benefit of on-line DSA is that the amount of analysis can be greatly reduced to include only those contingencies relevant to actual operating conditions.
From an engineering viewpoint, on-line security assessment requires evaluating the static as well as dynamic effects of hundreds or even thousands of credible contingencies on power systems. Static security assessment (SSA), now routinely performed in energy management systems, checks the degree of satisfaction for all relevant static constraints of post-fault (post-contingency) steady states. From a computational viewpoint, SSA needs to solve a large set. of nonlinear algebraic equations. Dynamic security assessment (DSA), concerned with power system stability/instability after contingencies, requires the handling of a large set of nonlinear differential equations in addition to the nonlinear algebraic equations involved in SSA. The computational effort required in on-line DSA is roughly three orders of magnitude higher than that for SSA.
To significantly reduce the computational burden required for on-line DSA, the strategy of using an effective scheme to screen out a large number of stable contingencies and to apply detailed simulation programs only to potentially unstable contingencies is well recognized. This strategy has been successfully implemented in on-line SSA and can potentially be applied to on-line DSA. Given a set of credible contingencies, the strategy would break the task of on-line DSA into two assessment stages:
Stage 1: Perform the task of fast dynamic contingency screening to screen out contingencies which are definitely stable from a set of credible contingencies
Stage 2: Perform a detailed stability assessment and energy margin calculation for each contingency remaining after Stage 1.
Dynamic contingency screening of Stage 1 is a fundamental function of an on-line DSA system. After Stage 1, the remaining contingencies, classified as undecided or potentially unstable, are then sent to Stage 2 for detailed stability assessment and energy margin calculation. Methods based on time-domain simulation can generally be applied to Stage 2 of on-line DSA. The overall computational speed of an on-line DSA system depends greatly on the effectiveness of the dynamic contingency screening, whose objective is to identify contingencies which are definitely stable and thus do not require further stability analysis.
Under the on-line application environment, the following five requirements are essential for any classifiers intended for on-line dynamic contingency screening of modern power systems:
(A-1) (reliability measure) the classifier absolutely captures unstable contingencies; specifically, no unstable (single-swing or multi-swing) contingencies can be missed by the classifier. In other words, the ratio of the number of captured unstable contingencies to the number of actual unstable contingencies is 1.
(A-2) (efficiency measure) the classifier achieves a high yield of screening out stable contingencies, i.e., the ratio of the number of stable contingencies screened out by the classifier to the number of actual stable contingencies is as close to 1 as possible.
(A-3) (on-line computation) the classifier has little need of off-line computations and/or adjustments in order to meet the constantly changing and uncertain operating conditions.
(A-4) (speed measure) high speed, i.e. the classifier is fast for the requirement of on-line operation.
(A-5) (performance measure) the performance of the classifier in DSA with respect to changes in power system operating conditions is robust.
The requirement of the absolute capture of unstable contingencies is a reliability measure for dynamic contingency screening. The requirement of a high percentage of stable contingency drop-outs is an efficiency measure. These measures should not be degraded for different operating conditions as dictated by the requirement of robust performance. The trend of current and future power system operating environments is that on-line operational data and presumed off-line data can be very different. In a not-too-extreme case, off-line presumed data may become uncorrelated with on-line operational data. This indicates the importance of the on-line computation requirement.
Several research developments in on-line dynamic contingency screening have been reported in the literature. At present, the existing methods for dynamic contingency screening, except the one discussed in [2,3], all rely tremendously on extensive off-line simulation results to classify contingencies. These screening methods all first perform extensive numerical simulation on a set of credible contingencies using off-line network data in order to capture essential stability features of system dynamical behaviors; they then construct a classifier attempting to correctly classify contingencies on new and unseen network data in an on-line mode. Hence, these methods cannot meet the above on-line computation requirements. Furthermore, these methods cannot meet the reliability requirement.
BCU Methods
Recently, a systematic method to find the controlling unstable equilibrium point, called the BCU method, was developed and is disclosed in U.S. Pat. No. 5,483,462 to Chiang [I]. In developing a BCU method for a given power system stability model, an associated reduced-state model must be defined first. We consider the general network-preserving transient stability model with losses shown below                     θ        =                                            -                                                ∂                  U                                                  ∂                  u                                                      ⁢                          (                              u                ,                w                ,                x                ,                y                            )                                +                                    g              1                        ⁡                          (                              u                ,                w                ,                x                ,                y                            )                                                              θ        =                                            -                                                ∂                  U                                                  ∂                  w                                                      ⁢                          (                              u                ,                w                ,                x                ,                y                            )                                +                                    g              2                        ⁡                          (                              u                ,                w                ,                x                ,                y                            )                                                                        T          ⁢                                           ⁢                      x            .                          =                                            -                                                ∂                  U                                                  ∂                  x                                                      ⁢                          (                              u                ,                w                ,                x                ,                y                            )                                +                                    g              3                        ⁡                          (                              u                ,                w                ,                x                ,                y                            )                                                                        y          .                =        z                                          M          ⁢                                           ⁢                      z            .                          =                                            -              D                        ⁢                                                   ⁢            z                    -                                                    ∂                U                                            ∂                y                                      ⁢                          (                              u                ,                w                ,                x                ,                y                            )                                +                                    g              4                        ⁡                          (                              u                ,                w                ,                x                ,                y                            )                                          where U(u, w, x, y) is a scalar function. Regarding the original model (1), we choose the following differential-algebraic system as the associated reduced-state model                     0        =                                            -                                                ∂                  U                                                  ∂                  u                                                      ⁢                          (                              u                ,                w                ,                x                ,                y                            )                                +                                    g              1                        ⁡                          (                              u                ,                w                ,                x                ,                y                            )                                                              0        =                                            -                                                ∂                  U                                                  ∂                  w                                                      ⁢                          (                              u                ,                w                ,                x                ,                y                            )                                +                                    g              2                        ⁡                          (                              u                ,                w                ,                x                ,                y                            )                                                                        T          ⁢                                           ⁢                      x            .                          =                                            -                                                ∂                  U                                                  ∂                  x                                                      ⁢                          (                              u                ,                w                ,                x                ,                y                            )                                +                                    g              3                        ⁡                          (                              u                ,                w                ,                x                ,                y                            )                                                                        y          .                =                                            -                                                ∂                  U                                                  ∂                  y                                                      ⁢                          (                              u                ,                w                ,                x                ,                y                            )                                +                                    g              4                        ⁡                          (                              u                ,                w                ,                x                ,                y                            )                                          
The fundamental ideas behind the BCU method can be explained as follows. Given a power system stability model (which admits an energy function), the BCU method first explores the special properties of the underlying model with the aim of defining an artificial, state-reduced model such that certain static as well as dynamic relationships are met. The BCU method then finds the controlling UEP of the state-reduced model by exploring the special structure of the stability boundary and the energy function of the state-reduced model. Finally, it relates the controlling UEP of the state-reduced model to the controlling UEP of the original model.
A Conceptual BCU Method
Step 1. From the fault-on trajectory (u(t), ω(t), x(t), y(t), z(t)) of the network-preserving model (1), detect the exit point (u*, w*, x*, y*) at which the projected trajectory (u(t), ω(t), x(t), y(t)) exits the stability boundary of the post-fault reduced-state model (2).
Step 2. Use the exit point (u*, w*, x*, y*), detected in Step 1, as the initial condition and integrate the post-fault reduced-state model to an equilibrium point. Let the solution be (uco, wco, xco, yco).
Step 3. The controlling UEP with respect to the fault-on trajectory of the original network-preserving model (1) is (uco, wco, xco, yco, 0). The energy function at (uco, wco, xco, yco, 0) is the critical energy for the fault-on trajectory (u(t), ω(t), x(t), y(t), z(t)).
Step 1 and Step 2 of the conceptual BCU method compute the controlling UEP of the reduced-state system. Note that the post-fault reduced-state trajectory stability from the exit point (u*, w*, x*, y*), Step 2 of the conceptual BCU method, will converge to an equilibrium point. Step 3 relates the controlling UEP of the reduced-state system (with respect to the projected fault-on trajectory) to the controlling UEP of the original system. There are several possible ways to numerically implement the conceptual BCU method for network-preserving power system models.
A numerical implementation of the conceptual BCU method for network-preserving power system models is presented below:
A Numerical BCU Method
Step 1. From the (sustained) fault-on trajectory (u(t), w(t), x(t), y(t),z(t)) of the original model (1), detect the exit point (u*, w*, x*, y*) at which the projected trajectory (u(t), w(t), x(t), y(t)) reaches the first local maximum of the numerical potential energy function.
Step 2. Use the point (u*, w*, x*, y*) as the initial condition and integrate the post-fault, reduced-state system (2) to the (first) local minimum of the following norm of the post-fault, reduced-state system (2). Let the local minimum be (u*0, w*0, x*0, y*0).
Step 3. Use the point (u*0, w*0, x*0, y*0) as the initial guess and solve the following set of nonlinear algebraic equations                                                               ∂              U                                      ∂              u                                ⁢                      (                          u              ,              w              ,              x              ,              y                        )                          +                              g            1                    ⁡                      (                          u              ,              w              ,              x              ,              y                        )                                      +                                                            ∂              U                                      ∂              w                                ⁢                      (                          u              ,              w              ,              x              ,              y                        )                          +                              g            2                    ⁡                      (                          u              ,              w              ,              x              ,              y                        )                                      +                                                            ∂              U                                      ∂              x                                ⁢                      (                          u              ,              w              ,              x              ,              y                        )                          +                              g            3                    ⁡                      (                          u              ,              w              ,              x              ,              y                        )                                      +                                                            ∂              U                                      ∂              y                                ⁢                      (                          u              ,              w              ,              x              ,              y                        )                          +                              g            4                    ⁡                      (                          u              ,              w              ,              x              ,              y                        )                                      =  0Let the solution be (u*co, w*co, x*co, y*co).
Step 4. The controlling u.e.p. relative to the fault-on trajectory (u(t), w(t), x(t), y(t), z(t)) of the original model is (u*co, w*co, x*co, y*co, 0)
Steps 1 to 3 of the above numerical network-preserving BCU method compute the control-line u.e.p. of the reduced-state system (2) and Step 4 relates the controlling u.e.p. of the reduced-state system to the controlling u.e.p. of the original system. In step 3 of the numerical BCU method, the minimum gradient point (MGP) is used as a guide to search for the controlling u.e.p. From a computational viewpoint, the MGP can be used as an initial guess in the Newton method to compute the controlling u.e.p. If the MGP is sufficiently close to the controlling u.e.p., then the sequence generated by the Newton method starting from the MGP will converge to the controlling u.e.p. Otherwise, the sequence may converge to another equilibrium point or diverge. A robust nonlinear algebraic solver should be used in Step 3.
BCU Classifiers
Recently, a set of BCU classifiers for the on-line dynamic contingency screening of electric power systems was developed [2,3], and is disclosed in U.S. Pat. No. 5,719,787 to Chiang and Wang [2]. However, the BCU classifiers may not always meet the five essential requirements as shown in the following numerical simulations.
Consider a 173-bus real power system model. A total of 1014 system contingencies with two different load models were screened using the BCU classifiers [1]. The types of fault considered in the evaluation were three-phase faults with fault locations at both generator and load buses. Some contingencies were faults which were cleared by opening double circuits while others were faults which were cleared by opening the single circuit. A ZIP load model with a composition of 20% constant current, 20% constant power and 60% constant impedance was used in the simulation. Both severe and mild faults were considered. All faults were assumed to have been cleared after 0.07 s. A reliable time-domain stability program was used to numerically verify all the classification results.
Giving a total of 507 contingencies to the BCU classifiers, the first BCU classifier dropped out 59 cases and classified them as unstable. These 59, cases were numerically verified by the time-domain stability program. Of the 59 cases, 58 cases were indeed unstable, according to the time-domain stability program, and 1 case was stable. The, remaining 448 contingencies were sent to the second BCU classifier for another classification. This classifier dropped 8 cases which were classified as stable and all of these were verified by the time-domain stability program as actually being either single-swing or multi-swing stable. Note that in practical application it is not necessary to send these stable contingencies (as classified by the BCU classifiers) to a time-domain program for verification. The remaining 440 contingencies were sent to BCU classifier III which screened out 0 unstable cases. The remaining 440 contingencies were sent to BCU classifier IV which screened out 332 stable cases. Among these, 10 cases were unstable, according to the time-domain stability program, and 322 cases were stable. The fifth BCU classifiers totally screened out 16 contingencies. Those contingencies were classified as unstable. Of these, 14 contingencies were stable and 2 were indeed unstable. The remaining contingencies entered the last BCU classifier for final classification. Among them, 0 cases were classified as stable, and 92 cases were classified as unstable. Among these, 12 cases were indeed unstable and 80 cases were stable, as verified by the time-domain stability program.
This numerical simulation reveals that the BCU classifiers may mis-classify unstable contingencies as stable. For instance, 10 unstable contingencies in the 173-bus system were mis-classified as stable, hence violating the reliability requirement of a dynamic security classifier.
This invention develops improved BCU classifiers for the on-line dynamical security screening of practical power systems. The improved BCU classifiers not only meet the five requirements described above but also make the strategy, previously in practice in static security assessments, applicable to on-line dynamical security assessments. Furthermore, improved BCU classifiers compute energy margins for screened stable contingencies.
To illustrate the effectiveness of the improved BCU classifiers in meeting the five essential requirements (A-1) through (A-5), we applied them to the 173-bus power system with the same system
TABLE 1The BCU classifiers on a 173-bus damped system: ZIP ModelIIIIIIIVVVIVIToolsResults(U)(S)(U)(S)(U)(S)(U)TotalBCUDrop-out598033216092507classifierscasesTime-Stable18032214080425DomainUnstable580010201282(ETMSP)system conditions and the same set of contingencies. The simulation results are presented below. Giving a total of 507 contingencies to the improved BCU classifiers, the first BCU classifier dropped out 83 cases and classified them as unstable. These 83 cases were numerically verified by the time-domain stability program. Among these, 74 cases were indeed unstable, according to the time-domain stability program, and 9 cases were stable. The remaining 424 contingencies were sent to the second BCU classifier for another classification. This classifier dropped 16 cases which were classified as stable, and they are indeed stable according to the time-domain stability program. The remaining 408 contingencies were sent to BCU classifier III which screened out 0 unstable cases. The remaining 408 contingencies were sent to BCU classifier IV which screened out 1 unstable case. This case, according to the time-domain stability program, was stable The fifth BCU classifier does not screen out any contingency. BCU classifier VI totally screened out 1 contingency which was classified as unstable. This contingency, however, is stable, according to the time-domain stability program. The remaining contingencies entered the last BCU classifier for final classification. Among them, 380 cases were classified as stable and all of these were verified by the time-domain stability program as stable; 26 cases were classified as unstable. Among these, 8 cases were indeed unstable and 18 cases were stable, as verified by the time-domain stability program.
This numerical simulation reveals that the improved BCU classifiers do not mis-classify unstable contingencies as stable, hence meeting the reliability requirement of a dynamic security classifier. We also applied the improved BCU classifiers to the 173-bus power system with the same set of contingencies and the same system conditions, except that the system dampings were set to zero. The simulation results are tabulated in Tables 2 & 3. Again, the improved BCU classifiers do not mis-classify unstable contingencies as stable on the test system.
Based on the above numerical simulations, we examine in the following the degree of satisfaction with which the improved BCU classifiers meet the essential requirements for performing on-line dynamic contingency screening of the 173-bus power system.
TABLE 2Improved BCU Classifiers on a 173-bus damped System: ZIP modelResults &IIIIII-AIII-BIVVVIVIIVIIToolsVerifications(U)(U)(S)(U)(U)(U)(U)(S)(U)TotalImproved BCUScreened0836020139025507ClassifierscasesTime-DomainStable066020139017422Unstable077000000885
TABLE 3Improved BCU Classifiers on a 173-bus undamped System: ZIP modelResults &IIIIII-AIII-BIVVVIVIIVIIToolsVerifications(U)(U)(S)(U)(U)(U)(U)(S)(U)TotalImproved BCUScreened08316010138026507ClassifierscasesTime-DomainStable0916010138018425(ETMSP)Unstable074000000882
TABLE 4Performance Evaluation of Improved BCU Classifiers ona 173-bus System: ZIP load model.#RequirementsDescriptionUndamped Damped1 AbsoluteNumber of captured unstable   100%  100%capture ofcases divided by the numberunstableof actual unstable casescontingenciesequals 1.02High yieldNumber of detected stable89.41%92.41%of stablecases divided by the numbercontingenciesof actual stable cases isclose to but less than 1.03Little off-lineIn compliance with on-lineYesYescomputationsrequirements. On-line dataand off-line data may havelittle correlation.4High speedFast classificationYesYes5RobustThe same threshold value ofYesYesperformanceeach classifier is appliedto different power systemoperating conditionsAbsolute Capture and Drop-out
The improved BCU classifiers meet the requirements of absolute capture of unstable contingencies on a total of 1014 contingencies. The capture ratio (i.e. the ratio of captured unstable contingencies to the actual contingencies) is 1.0. In other words, the improved BCU classifiers capture all of the unstable contingencies.
High Drop-out Stable Contingencies
The yield of drop-out (i.e. the ratio of the dropped-cut stable contingencies to the actual stable contingencies with the improved BCU classifiers) is 90.99% (damped). 90.58% (undamped), respectively. A summary of the reliability and efficiency measure of the improved BCU classifiers on these test systems is shown in Table 4. Note that the same threshold values for each of the eight BCU classifiers were applied to these 1014 cases. No off-line computation is required with the improved BCU classifiers.
BCU-guided Time-domain Method
We next turn to Stage 2 of the on-line DSA, which is involved with detailed stability assessment and energy margin calculation. After decades of research and developments in the direct methods, it has become clear that they can not replace the time-domain approach in stability analysis. Instead, the capabilities of direct methods and that of the time-domain approach complement each other. The current direction of development is to combine a direct method and a fast time-domain method into an integrated power system stability program to take advantages of the merits of both methods.
There are several direct methods proposed in the literature for computing energy margins. From a practical viewpoint, the existing direct methods can not reliably compute accurate energy margin for every contingency. Some direct methods can compute energy margin for just some type of contingencies while the other direct methods can compute energy margins for another type of contingencies. Hence, one has to resort to a time-domain based method for accurate energy margin calculation of those contingencies for which direct methods fail to compute. Indeed, the task of calculating an accurate energy margin for every contingency has long been regarded as a challenging one.
We propose that any time-domain based method intended for energy margin calculation must meet the following essential requirements:
(B-1) The critical energy values computed by the method must be accurate and reliable
(B-2) The critical energy values computed by the method must be compatible with the critical energy values computed by the controlling UEP
(B-3) The method must be reasonably fast.
One promising approach for developing such a time-domain based method is one based on a combination of a type of direct method and a few runs of time-domain simulation.
All of the existing time-domain based methods proposed thus far for computing energy margins are composed of the following two steps:
Step 1. (stability assessment) the time-domain approach is applied to simulate the system trajectory and then assess its stability based on the simulated post-fault trajectory.
Step 2. (energy margin calculation) the corresponding energy margin is calculated based on either the simulated post-fault trajectory alone (e.g. the equal-area criterion based methods and the hybrid method) or with the inclusion of some other system trajectories (e.g. the improved hybrid method and the second-kick method).
It is obvious that these methods discriminate stable and unstable contingencies very accurately for the model validity. They are, however, too slow for on-line applications and their accuracy in computing energy margins is not satisfactory. Moreover, these time-domain based methods cannot meet the requirements (B-1) through (B-3), stated above, mostly due to the following difficulties:                The critical energy value (hence the energy margin) can only be obtained after the critical clearing time is first calculated.        They lack a theoretical basis        The relationship between fault clearing time and energy margin is rather complex and may not be a functional relationship        For stable contingencies, the required computational time for time-domain simulation programs may be very long        
Hence, the existing time-domain methods for energy margin computation are not applicable to both Stage 1 and stage 2 of on-line DSA.
Recently, the second-kick method for computing the energy margin was disclosed in U.S. Pat. No. 5,638,297 to Mansour. Vaahedi and Chang [3]. However, as shown in several numerical simulations, the second-kick method cannot always meet essential requirements (B-1) through (B-3). In particular, the energy margins calculated by the second-kick method are usually incompatible and inconsistent with exact energy margins.
We believe that the only viable approach to develop a time-domain based method for computing energy margin is the one which satisfies the following guidelines:
(G-1) It is based on the calculation (or approximation) of the critical clearing time,
(G-2) It can effectively reduce the duration of the time interval within which time-domain stability simulations are performed in order to determine the critical clearing time. (Obviously, the shorter the duration of the time interval is, the lesser the number of time-domain stability simulations is required and the faster the method will be.)
The present invention develops a (two-stage) BCU-guided time-domain method, which is a time-domain based, BCU-guided method, for stability assessment and computing critical energy values. The method is reliable and yet fast for calculating energy margin whose value is compatible with that computed by the controlling UEP method. Hence, the method meets the essential requirements (B1) through (B3). The BCU-guided time-domain method uses a BCU-guided scheme to specify, within a given time interval, a reduced-duration time interval and employs the golden bisection interpolation algorithm to the specified time interval to reduce the total number of time-domain simulations required for accurate energy margin calculation.
We also develop in this invention a novel system, called BCU-DSA, for on-line dynamic security assessments and energy margin calculations of practical power systems in modern energy management systems. The novel system meets the requirements of on-line dynamic security assessment and energy margin calculations through effective exploration of the merits of both the BCU method (and the improved BCU classifiers) and the detailed time-domain simulation program. The architecture of the novel system is shown in FIG. 1. There are two major components in this architecture: (i) a dynamic contingency screening program made up of a sequence of improved BCU classifiers whose major functions are to screen out all of those contingencies which are definitely stable from a set of credible contingencies and to capture all of the (potentially) unstable contingencies, and (ii) a BCU-guided time-domain program for stability analysis and energy margin calculation of both the (potentially) unstable and undecided contingencies captured in (i).